Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve the inequality \( |1 - 6x| - 6 < -3 \), we need to simplify it step by step.
### Step 1: Isolate the Absolute Value
We start by isolating the absolute value expression:
[tex]\[ |1 - 6x| - 6 < -3 \][/tex]
Add 6 to both sides to isolate the absolute value:
[tex]\[ |1 - 6x| < 3 \][/tex]
### Step 2: Understand the Absolute Value Inequality
The inequality \( |1 - 6x| < 3 \) means that the expression inside the absolute value, \(1 - 6x\), is within 3 units of 0. This can be expressed as two separate inequalities:
[tex]\[ -3 < 1 - 6x < 3 \][/tex]
### Step 3: Break it into Two Inequalities
Now, we break this compound inequality into two separate inequalities:
1. \( 1 - 6x < 3 \)
2. \( 1 - 6x > -3 \)
### Step 4: Solve Each Inequality Individually
#### Inequality 1: \( 1 - 6x < 3 \)
Subtract 1 from both sides:
[tex]\[ -6x < 2 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x > -\frac{1}{3} \][/tex]
#### Inequality 2: \( 1 - 6x > -3 \)
Subtract 1 from both sides:
[tex]\[ -6x > -4 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x < \frac{2}{3} \][/tex]
### Step 5: Combine the Solutions
Combining the results from both inequalities, we get:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
### Graph the Solution Set
To graph the solution set, we focus on the interval \(-\frac{1}{3} < x < \frac{2}{3}\):
- Draw a number line.
- Mark the points \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line.
- Since the inequality does not include the endpoints, use open circles at \(-\frac{1}{3}\) and \(\frac{2}{3}\).
- Shade the region between \(-\frac{1}{3}\) and \(\frac{2}{3}\).
In summary, the solution to the inequality \( |1 - 6x| - 6 < -3 \) is:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
And the graph of the solution set is an open interval between \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line:
[tex]\[ \begin{array}{c} \text{ } \\ \underset{-1/3 \quad 0 \quad 2/3}{\overset{\ \ \ \ \ \small \bigcirc \quad \quad \small \bigcirc}{\line(1,0){150}}} \end{array} \][/tex]
### Step 1: Isolate the Absolute Value
We start by isolating the absolute value expression:
[tex]\[ |1 - 6x| - 6 < -3 \][/tex]
Add 6 to both sides to isolate the absolute value:
[tex]\[ |1 - 6x| < 3 \][/tex]
### Step 2: Understand the Absolute Value Inequality
The inequality \( |1 - 6x| < 3 \) means that the expression inside the absolute value, \(1 - 6x\), is within 3 units of 0. This can be expressed as two separate inequalities:
[tex]\[ -3 < 1 - 6x < 3 \][/tex]
### Step 3: Break it into Two Inequalities
Now, we break this compound inequality into two separate inequalities:
1. \( 1 - 6x < 3 \)
2. \( 1 - 6x > -3 \)
### Step 4: Solve Each Inequality Individually
#### Inequality 1: \( 1 - 6x < 3 \)
Subtract 1 from both sides:
[tex]\[ -6x < 2 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x > -\frac{1}{3} \][/tex]
#### Inequality 2: \( 1 - 6x > -3 \)
Subtract 1 from both sides:
[tex]\[ -6x > -4 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x < \frac{2}{3} \][/tex]
### Step 5: Combine the Solutions
Combining the results from both inequalities, we get:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
### Graph the Solution Set
To graph the solution set, we focus on the interval \(-\frac{1}{3} < x < \frac{2}{3}\):
- Draw a number line.
- Mark the points \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line.
- Since the inequality does not include the endpoints, use open circles at \(-\frac{1}{3}\) and \(\frac{2}{3}\).
- Shade the region between \(-\frac{1}{3}\) and \(\frac{2}{3}\).
In summary, the solution to the inequality \( |1 - 6x| - 6 < -3 \) is:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
And the graph of the solution set is an open interval between \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line:
[tex]\[ \begin{array}{c} \text{ } \\ \underset{-1/3 \quad 0 \quad 2/3}{\overset{\ \ \ \ \ \small \bigcirc \quad \quad \small \bigcirc}{\line(1,0){150}}} \end{array} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
